Properties

Label 360.2.a.a
Level 360
Weight 2
Character orbit 360.a
Self dual yes
Analytic conductor 2.875
Analytic rank 1
Dimension 1
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 360 = 2^{3} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 360.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(2.87461447277\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 40)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - q^{5} - 4q^{7} + O(q^{10}) \) \( q - q^{5} - 4q^{7} - 4q^{11} - 2q^{13} - 2q^{17} + 4q^{19} - 4q^{23} + q^{25} + 2q^{29} - 8q^{31} + 4q^{35} + 6q^{37} + 6q^{41} - 8q^{43} - 4q^{47} + 9q^{49} - 6q^{53} + 4q^{55} + 4q^{59} - 2q^{61} + 2q^{65} + 8q^{67} - 6q^{73} + 16q^{77} + 16q^{83} + 2q^{85} + 6q^{89} + 8q^{91} - 4q^{95} - 14q^{97} + O(q^{100}) \)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
0
0 0 0 −1.00000 0 −4.00000 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 360.2.a.a 1
3.b odd 2 1 40.2.a.a 1
4.b odd 2 1 720.2.a.e 1
5.b even 2 1 1800.2.a.v 1
5.c odd 4 2 1800.2.f.a 2
8.b even 2 1 2880.2.a.t 1
8.d odd 2 1 2880.2.a.bg 1
9.c even 3 2 3240.2.q.x 2
9.d odd 6 2 3240.2.q.k 2
12.b even 2 1 80.2.a.a 1
15.d odd 2 1 200.2.a.c 1
15.e even 4 2 200.2.c.b 2
20.d odd 2 1 3600.2.a.h 1
20.e even 4 2 3600.2.f.t 2
21.c even 2 1 1960.2.a.g 1
21.g even 6 2 1960.2.q.i 2
21.h odd 6 2 1960.2.q.h 2
24.f even 2 1 320.2.a.d 1
24.h odd 2 1 320.2.a.c 1
33.d even 2 1 4840.2.a.f 1
39.d odd 2 1 6760.2.a.i 1
48.i odd 4 2 1280.2.d.j 2
48.k even 4 2 1280.2.d.a 2
60.h even 2 1 400.2.a.e 1
60.l odd 4 2 400.2.c.d 2
84.h odd 2 1 3920.2.a.s 1
105.g even 2 1 9800.2.a.x 1
120.i odd 2 1 1600.2.a.o 1
120.m even 2 1 1600.2.a.k 1
120.q odd 4 2 1600.2.c.m 2
120.w even 4 2 1600.2.c.k 2
132.d odd 2 1 9680.2.a.q 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
40.2.a.a 1 3.b odd 2 1
80.2.a.a 1 12.b even 2 1
200.2.a.c 1 15.d odd 2 1
200.2.c.b 2 15.e even 4 2
320.2.a.c 1 24.h odd 2 1
320.2.a.d 1 24.f even 2 1
360.2.a.a 1 1.a even 1 1 trivial
400.2.a.e 1 60.h even 2 1
400.2.c.d 2 60.l odd 4 2
720.2.a.e 1 4.b odd 2 1
1280.2.d.a 2 48.k even 4 2
1280.2.d.j 2 48.i odd 4 2
1600.2.a.k 1 120.m even 2 1
1600.2.a.o 1 120.i odd 2 1
1600.2.c.k 2 120.w even 4 2
1600.2.c.m 2 120.q odd 4 2
1800.2.a.v 1 5.b even 2 1
1800.2.f.a 2 5.c odd 4 2
1960.2.a.g 1 21.c even 2 1
1960.2.q.h 2 21.h odd 6 2
1960.2.q.i 2 21.g even 6 2
2880.2.a.t 1 8.b even 2 1
2880.2.a.bg 1 8.d odd 2 1
3240.2.q.k 2 9.d odd 6 2
3240.2.q.x 2 9.c even 3 2
3600.2.a.h 1 20.d odd 2 1
3600.2.f.t 2 20.e even 4 2
3920.2.a.s 1 84.h odd 2 1
4840.2.a.f 1 33.d even 2 1
6760.2.a.i 1 39.d odd 2 1
9680.2.a.q 1 132.d odd 2 1
9800.2.a.x 1 105.g even 2 1

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(5\) \(1\)

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(360))\):

\( T_{7} + 4 \)
\( T_{11} + 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ \( 1 + T \)
$7$ \( 1 + 4 T + 7 T^{2} \)
$11$ \( 1 + 4 T + 11 T^{2} \)
$13$ \( 1 + 2 T + 13 T^{2} \)
$17$ \( 1 + 2 T + 17 T^{2} \)
$19$ \( 1 - 4 T + 19 T^{2} \)
$23$ \( 1 + 4 T + 23 T^{2} \)
$29$ \( 1 - 2 T + 29 T^{2} \)
$31$ \( 1 + 8 T + 31 T^{2} \)
$37$ \( 1 - 6 T + 37 T^{2} \)
$41$ \( 1 - 6 T + 41 T^{2} \)
$43$ \( 1 + 8 T + 43 T^{2} \)
$47$ \( 1 + 4 T + 47 T^{2} \)
$53$ \( 1 + 6 T + 53 T^{2} \)
$59$ \( 1 - 4 T + 59 T^{2} \)
$61$ \( 1 + 2 T + 61 T^{2} \)
$67$ \( 1 - 8 T + 67 T^{2} \)
$71$ \( 1 + 71 T^{2} \)
$73$ \( 1 + 6 T + 73 T^{2} \)
$79$ \( 1 + 79 T^{2} \)
$83$ \( 1 - 16 T + 83 T^{2} \)
$89$ \( 1 - 6 T + 89 T^{2} \)
$97$ \( 1 + 14 T + 97 T^{2} \)
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